Stochastic model for computer simulation of the number of cancer cells and lymphocytes in homogeneous sections of cancer tumors

We deal with a small enough tumor section to consider it homogeneous, such
that populations of lymphocytes and cancer cells are independent of spatial
coordinates. A stochastic model based in one step processes is developed to
take into account natural birth and death rates. Other rates are also
introduced to consider medical treatment: natural birth rate of lymphocytes and
cancer cells; induced death rate of cancer cells due to self-competition, and
other ones caused by the activated lymphocytes acting on cancer cells.
Additionally, a death rate of cancer cells due to induced apoptosis is
considered. Weakness due to the advance of sickness is considered by
introducing a lymphocytes death rate proportional to proliferation of cancer
cells.
Simulation is developed considering different combinations of the parameters
and its values, so that several strategies are taken into account to study the
effect of anti-angiogenic drugs as well the self-competition between cancer
cells. Immune response, with the presence of a kind of specialized lymphocytes,
is introduced such that they appear once cancer cells are detected. Induced
apoptosis of cancer cells is introduced to model the action of several drugs
under development right now. Besides, the model predicts the cancer relapse
even from a very small number of cells. Simulation is done by using Gillespie
algorithm.